Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations4 Marks
Question
Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$
✓

Answer

We have,
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2-\text{y}^2}{\text{xy}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x}^2-\text{v}^2\text{x}^2}{\text{vx}^2}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{1-\text{v}^2}{\text{v}}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{1-\text{v}^2}{\text{v}}-\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{1-2\text{v}^2}{\text{v}}$
$\Rightarrow\ \frac{\text{v}}{1-2\text{v}^2}\text{dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{\text{v}}{1-2\text{v}^2}\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \frac{-1}4\log\big|1-2\text{v}^2\big|=\log|\text{x}|+\log\text{C}$
$\Rightarrow\ \log\big|1-2\text{v}^2\big|=-4\log|\text{x}|-4\log\text{C}$
$\Rightarrow\ \log\big|\big(1-2\text{v}^2\big)\big(\text{x}^4\big)\big|=\log\frac{1}{\text{C}^4}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \log\big|\text{x}^2\big(\text{x}^2-2\text{y}^2\big)\big|=\log\frac{1}{\text{C}^4}$
$\Rightarrow\ \text{x}^2\big(\text{x}^2-2\text{y}^2\big)=\text{C}_1$
where
$\text{C}_1=\frac{1}{\text{C}^4}$
Hence, $\text{x}^2\big(\text{x}^2-2\text{y}^2\big)=\text{C}_1$ is the required solution.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be given as $a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k},~ b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$, and $c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}$ . Then show that $\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$
→
Evalute the following integrals:
$\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}$
→
If $\text{x}=\text{a}(\theta+\sin\theta)\ \text{and}\ \text{y}=\text{a}(1+\cos\theta)$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{a}}{\text{y}^2}.$
→
$\text{if y } = \sqrt{\frac{(x - 3)(x^{2} + 4)}{3x^{2} + 4x + 5}}, \text{find} \frac{dy}{dx}$
→
Solve the follwing system of equations by matrix method:
3x + 4y - 5 = 0
x - y + 3 = 0
→
Calculate the area of the ragion bounded by y2 = x and x2 = y.
→
Find the inverse of the following matrices by using elementry row transformation:

$\begin{bmatrix}7 & 1 \\4 & -3 \end{bmatrix}$

→
Find: $\int(3x + 1)\sqrt{4 - 3x - 2x^{2 } }\text{ dx}$
→
If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$
→
Let f be a function defined on [a, b] such that f ′(x) > 0, for all x $\in$ (a, b). Then prove that f is an increasing function on (a, b).
→